Initial ideals of tangent cones to Richardson varieties in the Orthogonal Grassmannian via a Orthogonal-Bounded-RSK-Correspondence

TL;DR

This paper describes the initial ideals of tangent cones to Richardson varieties in the Orthogonal Grassmannian using a new Orthogonal bounded RSK correspondence, extending previous results and providing explicit combinatorial descriptions.

Contribution

It introduces the Orthogonal bounded RSK correspondence and applies it to explicitly describe initial ideals of tangent cones in Orthogonal Grassmannian Richardson varieties.

Findings

Explicit initial ideals for tangent cones at T-fixed points.

A degree-preserving bijection between monomials and standard bases.

Generalization of RSK correspondence to the orthogonal setting.

Abstract

A Richardson variety $X_\ga^\gc$ in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety $X^\gc$ in the Orthogonal Grassmannian and a opposite Schubert variety $X_\ga$ therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any $T$-fixed point of $X_\ga^\gc$, thus generalizing a result of Raghavan-Upadhyay \cite{Ra-Up2}. Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal bounded RSK (OBRSK). The OBRSK correspondence will give a degree-preserving bijection between a set of monomials defined by the initial ideal of the ideal of the tangent cone (as mentioned above) and a `standard monomial basis'. A similar work for Richardson varieties in the ordinary Grassmannian was done by Kreiman in \cite{Kr-bkrs}.